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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> is <a href="rcomp_1.html#V3" target="_self">open</a>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a>
 ;<br/>

<b>assume </b><a NAME="E3:44"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1" target="_self">Neighbourhood</a> of  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:44"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">t</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:44"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2" target="_self">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font></span>)</span></span><a href="rcomp_1.html#K2" target="_self">.[</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:44"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E9:44"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<a NAME="E10:44"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E11:44"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<a NAME="E12:44"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E13:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:44"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


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